Given the profit function:

$P = -0.01x^2 + 176x - 100,000,$

we need to find the number of racquets, $x$, that the company should manufacture and sell to earn a profit of $606,800.

Step 1: Set up the equation

Set the profit equation equal to 606,800:

$-0.01x^2 + 176x - 100,000 = 606,800.$

Step 2: Rearrange the equation

Move all terms to one side to set the equation to zero:

$-0.01x^2 + 176x - 100,000 - 606,800 = 0,$

which simplifies to:

$-0.01x^2 + 176x - 706,800 = 0.$

Step 3: Solve the quadratic equation

The equation is in the standard form $ax^2 + bx + c = 0$, where:

• $a = -0.01$
• $b = 176$
• $c = -706,800$

We use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$

Substituting the values:

$x = \frac{-176 \pm \sqrt{(176)^2 - 4(-0.01)(-706,800)}}{2(-0.01)}.$

Let's compute the values inside the square root and solve for $x$.The solutions for the number of racquets the company should manufacture and sell to achieve a profit of $606,800 are:

$x = 6200 \quad \text{or} \quad x = 11400.$

Summary:

The company should manufacture and sell either 6,200 or 11,400 racquets.

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Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What is the maximum profit the company can achieve based on the profit function?
2. How many racquets should be produced to maximize the profit?
3. How would changes in the profit equation affect the break-even point?
4. What are the implications of having two solutions for this problem?
5. How does the discriminant affect the nature of the solutions?

Tip: When dealing with quadratic equations in real-world contexts, always consider the feasibility of each solution.